Lecture 4 - Inference Laws
Internal Argument
When we reason with each other in daily life, we make arguments.
An argument is a sequence of propositions, called hypotheses, followed by a final proposition, called conclusion.
Example:
If it is hot, I will wear shorts. (hypothesis)
It is hot. (hypothesis)
Therefore, I will wear shorts. (conclusion)
Internal Valid Argument
An argument is usually denoted as follows: p1 p2 …. pn ⊢ c (or c as the conclusion).
An argument is valid if the conclusion is true whenever the hypotheses are all true: (p1 \land p2 \land \dots \land p_n) \rightarrow c is a tautology.
An argument is invalid if the conclusion is false whenever the hypotheses are all true.
Key idea: validity concerns all possible truth assignments to the hypotheses, not actual truth in the real world.
Internal Example: Validity via Truth Table
Statement: If it is hot, I will wear shorts. (p → q)
Premises: p (It is hot)
Conclusion: q (I will wear shorts)
Task: Check if $((p \rightarrow q) \land p) \rightarrow q$ is a tautology.
Truth-table sketch (rows):
Row 1: p = T, q = T → p→q = T, (p→q) ∧ p = T, ((p→q) ∧ p) → q = T
Row 2: p = T, q = F → p→q = F, (p→q) ∧ p = F, ((p→q) ∧ p) → q = F
Row 3: p = F, q = T → p→q = T, (p→q) ∧ p = F, ((p→q) ∧ p) → q = T
Row 4: p = F, q = F → p→q = T, (p→q) ∧ p = F, ((p→q) ∧ p) → q = T
Conclusion: This argument is valid; a truth table (or logic laws) can establish validity.
Internal Is conclusion in a valid argument true?
Important point: A valid argument does not guarantee a true conclusion in the real world.
Only if all hypotheses of a valid argument are true, the conclusion is true.
Examples:
True example: All bears have fur. A polar bear is a bear. Therefore, polar bears have fur.
False example (premises not universally true): All bears live in the forest. A polar bear is a bear. Therefore, polar bears live in the forest.
takeaway: validity concerns form; truth of premises is a separate issue.
Internal Rules of Inference
Truth tables can be long and tedious.
Alternatively, we can prove arguments valid by applying rules of inference and laws of propositional logic in a logical proof.
Rules of inference are arguments that have been proved to be valid using truth tables.
Internal Rules of Inference: Common Rules
Modus ponens: If p is true and p → q is true, then q is true.
Symbolically: p,\; p \rightarrow q \vdash q
Modus tollens: If p → q is true and q is false, then p must also be false.
Symbolically: p \rightarrow q, \; \neg q \vdash \neg p
Hypothetical syllogism: If p → q and q → r, then p → r.
Symbolically: p \rightarrow q, \; q \rightarrow r \vdash p \rightarrow r
Disjunctive syllogism: If p ∨ q and ¬p, then q.
Symbolically: p \lor q, \; \neg p \vdash q
Addition: If p is true, then $p \lor q$ is also true (for any q).
Symbolically: p \vdash p \lor q
Simplification: If $p \land q$ is true, then p is true (and also q).
Symbolically: p \land q \vdash p \ p \land q \vdash q
Conjunction: If p is true and q is true, then $p \land q$ is true.
Symbolically: p, q \vdash p \land q
Resolution: From $p \lor q$ and ¬p, infer q (a form of eliminating a disjunctive option).
Symbolically: p \lor q, \; \neg p \vdash q
Note: The slide shows a table with these rules; the essence is the same standard set of propositional-inference rules. Some typography in the original may look garbled (e.g., a confusing description for Resolution), but the core ideas are these well-known rules.
Internal Write Logical Proof
A logical proof is a sequence of steps; each step has:
A proposition (the statement being asserted), and
A justification (the rule or hypothesis used to justify the step).
The proposition in any step can be a hypothesis or a statement proven by applying a rule to earlier steps.
The justification can be either a hypothesis or the rule applied.
The last step’s proposition must be the conclusion of the argument.
Internal Example: Proof Sequence
Given:
Proposition (p ∨ r) → q
q → t
r
Goal: derive t.
Steps:
r — Hypothesis
p ∨ r — Addition, from 1
(p ∨ r) → q — Hypothesis
q — Modus ponens, from 2 and 3
q → t — Hypothesis
t — Modus ponens, from 4 and 5
Result: t is derived using a short chain of inferences.
Internal Example: Hard Workout and Aspirin
Propositions:
w: I have a hard workout
s: I am sore
a: I take Aspirin
Rules:
w → s
s → a
Additional fact: ¬a (I am not taking Aspirin)
Also given: ¬w (I didn’t workout)
How it plays out:
From w → s and w is not assumed, but s → a and ¬a imply ¬s (modus tollens on s → a with ¬a).
From ¬s and w → s, you can infer ¬w (modus tollens with w → s and ¬s).
This sequence shows how the premises lead to ¬w, aligning with the explicit statement “I didn’t workout.”
Internal Example: More Complex Proof (Disjunction, Conjunctions, and Syllogisms)
Given:
p (q t) ¬u → ¬t ¬u q (note: the exact notation here is from the transcript; it shows a sequence intended to demonstrate multiple rule applications)
Step-by-step:
¬u → ¬t — Hypothesis
¬u — Hypothesis
¬t — Modus Ponens, 1, 2
p (q t) — Hypothesis
q t — Simplification, 4
t q — Commutative law, 5
q — Disjunctive syllogism
Note: The line-up appears to combine disjunctions and simplifications to derive q; some notation in the slide is not perfectly standard, but the intended flow is to show a multi-step proof using standard rules.
Internal Reasoning of Quantified Statements
Rules of reference do not always work with quantified statements.
Let the domain be the students in this class.
P(x): x got an A.
Q(x): x worked hard.
Question: Is the following argument valid?
∀x (P(x) ∨ Q(x))
∃x ¬P(x)
Therefore, ∃x Q(x)
Intuition: If for all x , either P(x) or Q(x) holds, and there exists some x with ¬P(x) , then that same x must satisfy Q(x) (since P(x) is false for that x ). Hence ∃xQ(x).
Quantified Reasoning: Instantiation and Generalization Rules
Existential Instantiation (EI): To use rules with a quantified statement, replace the quantified variable with a specific element from the domain.
If something is true for all elements in a domain, then it is true for any particular element c.
Universal Instantiation (UI): If there exists at least one element with a property, you may introduce a constant c such that P(c) is true (we don’t know which element, just that one exists).
Existential Generalization (EG): If a property holds for some specific element c, conclude that there exists at least one element with that property.
Universal Generalization (UG): If you can prove that P holds for an arbitrary element c, then it holds for all elements in the domain.
Internal Example: Quantified Argument (Domain = students in a class)
Prove the following argument is valid:
Everyone studied hard or failed the exam (or both).
Someone did not study hard.
Therefore, someone failed the exam.
Formalization:
P(x): x studied hard.
Q(x): x failed the class.
∀x (P(x) ∨ Q(x))
∃x ¬P(x)
Therefore, ∃x Q(x)
Internal Example 1.x: Existential Instantiation and Generalization in Practice
Steps:
¬P(x) — Hypothesis
(c is a specific student in the class) ¬P(c) — Existential Instantiation, 1
c is a specific student in the class — Simplification, 2
x (P(x) ∨ Q(x)) — Hypothesis
P(c) ∨ Q(c) — Universal Instantiation, 3, 4
¬P(c) — Simplification, 2
Q(c) — Disjunctive Syllogism, 5, 6
x Q(x) — Existential Generalization, 3, 7
Therefore,
x (P(x) ∨ Q(x))
x ¬P(x)
Therefore, x Q(x)
This sequence demonstrates how to move from quantified premises to a quantified conclusion via EI, UI, EG, and DG (disjunctive syllogism) reasoning.